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A Bayesian approach to decision making in early development trials

Audrey Yeo
M Nursing (Sydney), M Sci Biostats (Zürich)
Bristol Myers Squibbs 2025

This Quarto-rendered presentation has ALT text and as much as possible, uses colour-blind friendly palettes

Agenda

• Introduction & Motivation for Predictive Analytics at GPS
• Introducing phase1b
  • Foundations of Beta-Binomial Posterior
  • Two types of Posteriors
  • Monte Carlo Simulation / Operating Characteristics
  • What’s this for ?

My Journey so far

Career

• Health Care/ Academia 2018-2020

  -   Uni of Zürich : 
      - TA for Probability Theory I 
      - RA for Center for Reproducible Science 
      - RA Development Econometrics
  
  -   Royal Children's Hospital
      - Registered Nurse, COVID ICU/ Neonatal ICU

• joined Market Access at Roche mid 2021

  - RWD Enabling platform (Statistical Computing Environment, RWD assets, SQL)

Career

• joined R & D at Roche in mid 2022 :

  - Biostats Lead in Early Oncology Trials
  - Study Statistician for phase I-II, phase III trials
  - Led Roche/Genentech Dose Escalation on new SCE
  - Led the development of R package phase1b
  - Instructor for Julia Course Basel (Data Science, Quarto, ML modules)
  - Co-organiser and Presenter many Methodology seminars

• phase1b’s world debut in 2024 :

  - PSI 2024
  - useR!2024 
  - R/Pharma 2024
  - Effective Statistician conference 
  - BMS, UBC, Roche/ Genentech

Analysts are Strategic partners in the Pharma business :

… for therapeutics, we understand the combined Scientific and Business development of our assets such as :

• Roadblocks, business assumptions and strategies on development
• Scientific profiles of our assets e.g. half-life
• Target audience, regular and edge cases for use
• Regulatory frameworks

Developing phase1b has strengthened my understanding of this Strategic Partnership with target audience

About phase1b

• 2015 : Started as a need in Roche’s early development group, package development led by Daniel Sabanés Bové in 2015.

• 2023 : Refactoring, Renaming, adding Unit and Integration tests as current State-of-Art Software Engineering practice.

• 100% written in R and Open Source.

• website : genentech.github.io/phase1b/

library(devtools)
devtools::install_github("https://github.com/Genentech/phase1b")
library(phase1b)

Use case:

A single arm novel therapeutic with an assumed control response rate is at most 60%

Example	Interim	Final
Responders	16	23
n	23	40
Response rate	69.57 %	57.5 %
Posterior probability*	ask phase1b	ask phase1b
Predictive posterior probability*	ask phase1b	-
Decision to develop molecule further : Go/Stop/Grey Zone	ask phase1b	ask phase1b

Prior and Posterior of Beta Distribution for \(\pi\)

• Conjugate Prior is \(f(\pi)\), where \(\pi \sim {Beta(\alpha, \beta)}\), same family of distribution of Posterior
• We know the mean response rate (RR) is : \[\pi = \ \frac {\alpha}{\alpha + \beta}\]
• Likelihood is \(f(x|\pi)\), where \(x \sim {Binomial(x, n)}\)

• The updated Posterior \(f( \pi | x )\) is again a \(Beta\) distribution (same family as prior) : \[ \pi| \ x \sim Beta(\alpha + x, \ \beta + n - x)\] where \(x\) is the number of responders of current trial
• Posterior Probability :
  \(P (\pi > 60 \% | \alpha + x, \beta + n - x )\)
• Predictive Posterior Probability :
  \(P (success \ or \ failure \ at \ final)\)

Posterior formulation :

\[ f(\pi | x) \propto \ \pi^{x} (1-\pi)^{n-x} * \frac {1}{B(\alpha, \beta)} \pi^{\alpha-1}(1-\pi)^{\beta-1} \]

• (weighted sum version)

\[ f(\pi | x) \propto \ \pi^{x} (1-\pi)^{n-x} * \sum_{j = 1}^{k} \ w_j \frac {1}{B(\alpha_j, \beta_j)} \pi^{\alpha_j-1}(1-\pi)^{\beta_j-1} \]

Updating the Posterior

Using the formula for the mean, mode and median, where :

• \(\alpha = 0.6, \beta = 0.4\) and

• interim x = 16, interim n = 23 :
  \[ \pi = \ \frac {\alpha}{\alpha + \beta} = \ \frac {\alpha_{updated} }{\alpha_{updated} + \beta_{updated}} = \ \frac {16.6 }{16.6 + 7.4} ≈ 69.17 \% \]

  \[ mode (\pi) = \ \frac {\alpha_{updated} -1 }{\alpha_{updated} + \beta_{updated} - 2} = \ \frac {16.6 -1 }{16.6 + 7.4 - 2} ≈ 70.90 \% \] \[ median ( \pi) ≈ \ \frac {\alpha_{updated}}{ \alpha_{updated} + \beta_{updated} - \frac{2}{3} } ≈ 69.71 \% \]

Graphical representation of the updated Posterior

• Prior parameters are \(\alpha\) = 0.6, \(\beta\) = 0.4
• Updated Posterior parameters are \(\alpha\) = 16.6 and \(\beta\) = 7.4

Effect on varying weight on different priors Example

A variety of Priors

• To illustrate how density of Prior changes with increased sample size even though mean is the same

A variety of Posteriors

• To illustrate how density of Posterior changes with increased sample size even though mean is the same

postprob() example (Lee & Liu, 2008)

Example	Interim
Responders	16
n	23
Response rate	69.57 %
Standard of Care Response rate	60 %
Posterior probability	postprob( ) call from phase1b

phase1b::postprob(x = 16, n = 23, p = 0.60, par = c(0.6, 0.4))

[1] 0.8359808

Posterior Probability

• Interim trial is efficacious if posterior probability exceeds 70% or P( RR ≥ 60 % | data ) > 70%

Figure 1

Beta prior mixture

• phase1b facilitates the flexibility of using various priors and its respective weightings:

             Prior is P_E ~ sum(weights * beta(parE[, 1], parE[, 2]))

a = phase1b::postprob(x = 16,
             n = 23,
             p = 0.60,
             parE = c(0.6, 0.4), weights = 1)

b = phase1b::postprob(x = 16,
             n = 23,
             p = 0.60,
             parE = c(2, 4), weights = 1)

0.5*(a + b)

phase1b::postprob(x = 16,
             n = 23,
             p = 0.60,
             parE = rbind(c(0.6, 0.4), c(2, 4)), weights = c(0.5, 0.5))

• Posterior formulation :

\[ f(\pi | x) \propto \ \pi^{x} (1-\pi)^{n-x}\sum_{j = 1}^{k} \ w_j \frac {1}{B(\alpha_j, \beta_j)} \pi^{\alpha_j-1}(1-\pi)^{\beta_j-1} \]

predprob() example (Lee & Liu, 2008)

Example	Interim
Responders	16
n	23
Response rate	69.57 %
Standard of Care Response rate	60 %
Predictive Posterior probability	predprob( ) call from phase1b

control = 0.6
confidence_seventy = 0.7
result <- phase1b::predprob(
  x = 16, n = 23, Nmax = 40, p = control, thetaT = confidence_seventy,
  parE = c(0.6, 0.4)
)
result$result

[1] 0.8211011

confidence_ninety = 0.9
result_high_thetaT <- phase1b::predprob(
  x = 16, n = 23, Nmax = 40, p = control, thetaT = confidence_ninety,
  parE = c(0.6, 0.4)
)
result_high_thetaT$result

[1] 0.5655589

Predictive Posterior Probability

Predictive Posterior Probability is the Posterior probability of additional responders.

        (Note : 40 - 23 = 17 remaining patients. Potentially 16 + 17 responders at final = 33)

(a) \(P (\pi > 0.6 | \ data )\) > 70%

(b) \(P (\pi > 0.6 | \ data )\) > 90%

Figure 2: Predictive Posterior CDF for different Efficacy Rules

Operating Characteristics : Monte Carlo Simulations and threshold for Success (and Failure)

• Efficacy criteria, e.g. we would stop for Efficacy (Success) if : Pr( RR > p1) > tU
• Futility criteria, eg. we would stop for Futility (Failure) if : Pr( RR < p0) > tL

Rules and Operating characteristics. A use case for ocPostprob():

• Stop for Efficacy: Go if \(P( \pi > 60\% | \ data ) > 90 \%\)
• Stop for Futility: Stop if \(P( \pi < 60\% | \ data ) > 70 \%\)
• Prior of treatment arm \(Beta(0.6, 0.4)\).

set.seed(2025)
res <- phase1b::ocPostprob(
  nnE = c(23, 40), 
  truep = 0.60, 
  p0 = 0.60, 
  p1 = 0.60, 
  tL = 0.70, 
  tU = 0.90, 
  parE = c(0.6, 0.4),
  sim = 500, 
  wiggle = TRUE, 
  nnF = c(23, 40)
)

Results

ExpectedN	PrStopEarly	PrEarlyEff	PrEarlyFut	PrEfficacy	PrFutility	PrGrayZone
34	36.8 %	8.8 %	28 %	11 %	40 %	49 %

• At n = 34, we have some useful results (we did not have to evaluate 40!)

• 1/3 of results are known at interim, most of these results are in the Gray Zone

• ~ 1/2 of results are also at the Gray Zone at final

Expanded features

…. and wiggle room!

	SOC uncertainty	single-arm	two-arm	simulation	plotting	boundaries
postprob		✔️
postprobDist	✔️	✔️
predprob		✔️
predprobDist	✔️	✔️
ocPostprob		✔️		✔️
ocPostprobDist	✔️	✔️		✔️
ocPredprob		✔️		✔️
ocPredprobDist	✔️	✔️		✔️
ocRctPostprobDist	✔️	✔️	✔️	✔️
ocRctPredprobDist	✔️	✔️	✔️	✔️
plotBeta				✔️	✔️
plotDecision					✔️
plotOc					✔️
plotBounds					✔️
boundsPostprob						✔️
boundsPredprob						✔️

Some references

• Thall P F, Simon R (1994), Practical Guidelines for Phase IIB Clinical Trials, Biometrics, 50, 337-349

• Lee J J, Liu D D (2008), A Predictive probability design for phase II cancer clinical trials, 5(2), 93-106, Clinical Trials

• Yeo, A T, Sabanés Bové D, Elze M, Pourmohamad T, Zhu J, Lymp J, Teterina A (2024). Phase1b : Calculations for decisions on Phase 1b clinical trials. R package version 1.0.0, https://genentech.github.io/phase1b

• Inclusive Speaker Orientation Linux Foundation

• Zeileis, Fisher, Hornik, Ihaka, McWhite, Murrell, Stauffer, Wilke (2020) colorspace: A Toolbox for Manipulating and Assessing Colors and Palettes. Journal of Statistical Software.

Thanks

Audrey Yeo
M Nursing (Sydney), M Sci Biostats (Zürich)
Bristol Myers Squibbs

• Code for this talk
• Visit my website
• LinkedIn
• My GitHub repo

Outlook 2025

• DAGStat Conference presentation on Bayesian Statistics

• ISCB pre-course on Good Software Engineering practices

• R/Pharma Summit Board

• RoES conference submission

• CRAN submission for phase1b

• Improve Bouldering Grade to V6+

Effect on varying weight on different priors Example 2

State-of-Art R software/packages are currently valuable because it makes robust analysis, faster. One example is the phase1b

In a nutshell :
A robust, reliable and reproducible tool that calculates the Bayesian posterior probability and predictive posterior probability.

What is a state-of-art tool ? Check out a past workshop here

• Integrative and Unit testing
• Example and good documentation
• Assertions on input arguments and internal results that get passed on
• Check expected and actual results

Talks 2025

Year	Month	Date	Firm	Talk
2025	March	24	DAGStat	Practical Bayesian Statistics : A gentle refresher on probability theory, Bayesian Framework and Intuition for effective application in Biostatistics.
2025	March	24	DAGStat	Good Software Engineering Practice for R Packages

Associations

Firm	Title	Since
Swiss Statistical Society	Member	2020
openstatsware	Member	2024
R/Pharma	Organization committee	2024
open Source in Pharma's Diversity Alliance	Member	2024
EFSPI	event Volunteer	2024